1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2 17. ENTROPY, μ,-INVARIANT, AND FINITE TIME SINGULARITIES

1.1. Perelman's entropy and its associated invariants.

In this subsection we recall some basic facts regarding Perelman's energy

and entropy functionals, including the logarithmic Sobolev inequality, which

shall be used in this chapter.

1.1.1. Energy, entropy, and their minimizers.

Let g be a C^00 Riemannian metric on a closed manifold Mn, let J : M --+

~be a C^00 function, and let T E (0, oo). Perelman's energy functional is

defined by (see (5.1) in Part I)

(17.1) F (g, J) ~JM ( R + l\7 Jl^2 ) e-f dμ =JM (R + b.J) e-f dμ.

We may rewrite F as

(17.2) F(g,f)= jM(Rv^2 +4l\7vl^2 )dμ~Q(g,v),

where v ~ e-f/^2. The associated >.-invariant is (see (5.45) in Part I)

(17.3) >.(g) ~inf {F(g,J): J E C^00 (M), JM e-fdμ= 1}

=inf { Q(g,v): JM v^2 dμ = 1}.

There exists a unique C^00 minimizer Jo of F(g, J) subject to the constraint

JM e-f dμ = 1. Moreover, Jo satisfies the Euler-Lagrange equation (see

Lemma 5.23 in Part I)

(17.4) 2b.Jo -1\7 Jol^2 + R = >. (g).

Perelman's entropy functional Wis given by (see (6.1) in Part I)

(17.5)

where

(17.6)

We may rewrite W as

W(g,J,T)= r (T(Rw2+4l\7wl2) )dμ

JM -(log (w^2 ) + ~ log(4nT) + n) w^2

(17.7) ~K(g,w,T).

Recall that the associated μ-invariant is defined by (see (6.49) in Part I)

(17.8) μ(g,T) ~inf { W(g,J,T): J E C^00 (M), JM udμ = 1}

(17.9) =inf { K (g, w, T) : JM w^2 dμ = 1}.